Lyapunov exponents for Quantum Channels:an entropy formula and generic properties

Jader E. Brasil, Josue Knorst and Artur O. Lopes

August 23, 2019

Abstract

We denote by Mk the set of k by k matrices with complex entries. Weconsider quantum channels φL of the form: given a measurable functionL : Mk → Mk and a measure µ on Mk we define the linear operator φL :Mk →Mk, by the law ρ → φL(ρ) =

∫MkL(v)ρL(v)† dµ(v).

On a previous work the authors show that for a fixed measure µ it isgeneric on the function L the Φ-Erg property (also irreducibility). Here wewill show that the purification property is also generic on L for a fixed µ.

Given L and µ there are two related stochastic process: one takes valueson the projective space P (Ck) and the other on matrices in Mk. The Φ-Ergproperty and the purification condition are good hypothesis for the discretetime evolution given by the natural transition probability. In this way itwill follow that generically on L, if

∫|L(v)|2 log |L(v)| dµ(v) <∞, then the

Lyapunov exponents ∞ > γ1 ≥ γ2 ≥ ... ≥ γk ≥ −∞ are well defined.On the previous work it was presented the concepts of entropy of a channel

and of Gibbs channel; and also an example (associated to a stationary Markovchain) where this definition of entropy (for a quantum channel) matchesthe Kolmogorov-Shanon definition of entropy. We estimate here the largerLyapunov exponent for the above mentioned example and we show that it isequal to −1

2h, where h is the entropy of the associated Markov probability.

1 Introduction

We consider quantum channels of the form φL : Mk → Mk, where Mk isthe set of complex k by k matrices, ρ → φL(ρ) =

∫MkL(v)ρL(v)† dµ(v),

L : Mk →Mk is a measurable function and µ is a measure on Mk.In the recent paper [10] the authors considered Lyapunov exponents for

this class of channels φL when L was constant and equal to the identity

1

matrix. The Φ-Erg property and the purification condition (see definitionson section 6) were also considered on this mentioned paper.

On a previous paper [12] we show that for a fixed measure µ it is genericon the function L the Φ-Erg property (in fact we show that the irreduciblecondition is generic). The novelty here is that we will show that the pu-rification condition is also generic on L for a fixed measure µ (see section9).

The introduction of this variable L allows us to consider questions ofgeneric nature in this type of problem. We use the C0 topology in the set ofcomplex matrices.

Following [10] one can consider associated to L and µ two related process:one denoted by Xn, n ∈ N, takes values on the projective space P (Ck), andthe other, denoted by ρn, n ∈ N, takes values on Dk (where Dk is the set ofdensity operators). The natural transition probability is defined in [10].

The Φ-Erg property and the purification property play an important rolewhen analyzing the ergodic properties of these two processes.

For a fixed µ and a general L it was presented in [12] a natural conceptof entropy (see future section 3) for a channel in order to develop a versionof Gibbs formalism. It was also presented in example 8.5 in [12] a certainchannel (related to stationary Markov Chains) where the value obtained withthis definition coincides with the classical value of entropy. This shows thatthe concept that was introduced is natural. This definition of entropy is ageneralization of the concept described on the papers [6], [8] and [7]. Thisparticular way of defining entropy is in some sense inspired by results of [26]which considers iterated function systems.

The main contribution to the topic of Lyapunov exponents of quantumchannels is the paper [10]. We adapt here the formalism of [10] to the caseof a general L in order to estimate the Lyapunov exponents of the associateddynamical time evolution. We will just outline the proof (see section 7)because is basically the one in [10]). We describe the sufficient conditionsfor the Lyapunov exponents to be finite. Irreducibility and the purificationcondition are the good assumptions. Proposition 9.1 and [12] show that theseconditions are true for a fixed µ and a generic L.

Results relating entropy and Lyapunov exponents (both in the classicalsense) are quite important in Ergodic Theory (see for instance [9], [20] and[18]).

Another important issue here is the entropy formula. We compute thefirst Lyapunov exponent (which is negative) for the above mentioned example(see Section 8) and we show that it is equal to −1

2h, where h is the entropy of

the associated Markov probability. We also show that the second Lyapunovexponent in this case will be −∞. Of course, a general result for the class of

2

all quantum channels is not reachable due to its inherent generality.We point out that the definition of entropy for a (normalized) channel

presented in [12] explore the use of a “kind” of Ruelle operator. This proce-dure uses a natural a priori probability and this makes sense due to the factthat the “set of preimages” can be an uncountable set (see [19]). The mainissue on the reasoning in [12] is invariance (in time one), however, there theconcept of entropy is not directly associated to time evolution. We do notuse in this way ergodicity (or, the limit of measures of an increasing familyof partitions, etc..) in the definition of entropy. On the other hand, we pointout that the values of the Lyapunov exponents are of dynamical nature. Thisdynamical discrete time evolution is described by a stochastic process takingvalues on the set of matrices in Mk (see sections 6 and 7). The example weconsider here in section 8 shows that the concept of entropy of a channel (atleast in this case) presented in [12] can be linked to the natural dynamicaltime evolution via the main Lyapunov exponent.

Nice references for Quantum Channels are [16], [23], [1] and [29]. Thebook [28] presents several important results for the general theory of Lya-punov exponents (see also [14], [2], [3], [4], [5], [17] and [13]). [27] and [22]describe basic result in Ergodic Theory.

We thanks S. Klein for supplying us with references.

2 Basic results

We denote by Mk the set of complex k by k matrices. We denote by Id theidentity matrix on Mk.

We consider the standard Borel sigma-algebra over Mk and the canonicalEuclidean inner product on Ck

According to our notation † denotes the operation of taking the dual ofa matrix with respect to the canonical inner product on Ck.

Here tr denotes the trace of a matrix.Given two matrices A and B we define the Hilbert-Schmidt product

〈A , B 〉 = tr (AB†).

This induces a norm ‖A‖ =√〈A , A 〉 on the Hilbert space Mk which

will be called the Hilbert-Schmidt norm.

Definition 2.1 Given a linear operator Φ on Mk we denote by Φ∗ : Mk →Mk the dual linear operator in the sense of Hilbert-Schmidt, that is, if for allX, Y we get

〈Φ(X) , Y 〉 = 〈X , Φ∗(Y ) 〉.

3

Consider a measure µ on the Borel sigma-algebra over Mk. For an inte-grable transformation F : Mk →Mk:∫

Mk

F (v) dµ(v) =

(∫Mk

F (v)i,j dµ(v)

)i,j

,

where F (v)i,j is the entry (i, j) of the matrix F (v).

Definition 2.2 Given a measure µ on Mk and a measurable funtion L :Mk →Mk, we say that µ is L-square integrable, if∫

Mk

‖L(v)‖2 dµ(v) <∞.

For a fixed L we denote byM(L) the set of L-square integrable measures.We also denote P(L) the set of L-square integrable probabilities.

φL is well defined for L ∈M(L).

Proposition 2.3 Given a measurable function L : Mk → Mk and a squareintegrable measure µ, then, the dual transformation φ∗L is given by

φ∗L(ρ) =

∫Mk

L(v)†ρL(v) dµ(v).

Definition 2.4 Given a measure µ over Mk and a square integrable trans-formation L : Mk → Mk we say that L is a stochastic square integrabletransformation if

φ∗L(Id) =

∫Mk

L(v)†L(v) dµ(v) = Id.

Definition 2.5 A linear map φ : Mk → Mk is called positive if takespositive matrices to positive matrices.

Definition 2.6 A positive linear map φ : Mk → Mk is called completelypositive, if for any m, the linear map φm = φ⊗ Im : Mk⊗Mm →Mk⊗Mm

is positive, where Im is the identity operator acting on the matrices in Mm.

Definition 2.7 If φ : Mk →Mk is square integrable and satisfies

1. φ is completely positive;

4

2. φ preserves trace,

then, we say that φ is a quantum channel.

Theorem 2.8 Given µ and L square integrable then the associated trans-formation φL is completely positive. Moreover, if φL is stochastic, then itpreserves trace.

For the proof see [12].

Remark 2.9 φ∗L is also completely positive. We say that φL preserves unityif φL(Idk) = Idk. In this case, φ∗L preserves trace. If φ∗L preserves the identitythen φL preserves trace.

Definition 2.10 (Irreducibility) We say that φ : Mk → Mk is an irre-ducible channel if one of the equivalent properties is true

• Does not exists λ > 0 and a projection p in a proper subspace of Ck,such that, φ(p) ≤ λp;

• For all non null A ≥ 0, (1 + φ)k−1(A) > 0;

• For all non null A ≥ 0 there exists tA > 0, such that, (etAφ)(A) > 0;

• For all pair of non null positive matrices A,B ∈ Mk there exists anatural number n ∈ 1, ..., k − 1, such that, tr [Bφn(A)] > 0.

For the proof of the equivalences we refer the reader to [15], [25] and [29].

Definition 2.11 (Irreducibility) Given µ we will say (by abuse of lan-guage) that L is irreducible if the associated φL is an irreducible channel.

Theorem 2.12 (Spectral radius of φL and φ∗L) Given a square integrableL : Mk →Mk assume that the associated φL is irreducible. Then, the spectralradius λL > 0 of φL and φ∗L is the same and the eigenvalue is simple. Wedenote, respectively, by ρL > 0 and σL > 0, the eigenmatrices, such that,φL(ρL) = λLρL and φ∗L(σL) = λLσL, where ρL and σL are the unique nonnull eigenmatrices (up to multiplication by scalar).

The above theorem is the natural version of the Perron-Frobenius Theo-rem for the present setting.

It is natural to think that φL acts on density states and φ∗L acts in self-adjoint matrices.

5

Definition 2.13 Given the measure µ over Mk we denote by L(µ) the setof all integrable L such that the associated φL is irreducible.

Definition 2.14 Suppose L is in L(µ). We say that L is normalized if φLhas spectral radius 1 and preserves trace. We denote by N(µ) the set of allnormalized L.

If L ∈ N(µ), then, we get from Theorem 2.12 and the fact that φ∗L(Idk) =Idk, that λL = 1. That is, there exists ρL such that φL(ρL) = ρL and ρL isthe only fixed point. Moreover, the spectral radius is equal to 1.

Theorem 2.15 (Ergodicity and temporal means) Suppose L ∈ N(µ).Then, for all density matrix ρ ∈Mk it is true that

limN→∞

1

N

N∑n=1

φnL(ρ) = ρL,

where ρL is the density matrix associated to L.

Proof: The proof follows from Theorem 2.12 and corollary 6.3 in [29].

The above result connects irreducibility and ergodicity (the temporalmeans have a unique limit).

3 Entropy

A measure µ over Mk (which plays the role of the a priori probability) isfixed. In this way given L ∈ L(µ) we will associate in a natural way thetransformation φL : Mk →Mk.

Definition 3.1 We denote by Φ = Φµ the set of all L such that the associatedφL : Mk →Mk is irreducible and stochastic.

Suppose L is irreducible and stochastic.Given L consider the density matrix ρL which is invariant for φL (see

Theorem 2.12).

Definition 3.2 We define entropy for L (or, for φL) by the expression (whenfinite) :

6

h(L) = hµ(L) := −∫Mk×Mk

tr (L(v)ρLL(v)†)PL(v, w) logPL(v, w) dµ(v)dµ(w),

where

PL(v, w) :=tr (L(w)L(v)ρLL(v)†L(w)†)

tr (L(v)ρLL(v)†).

This definition is a generalization of the analogous concept presented onthe papers [6], [8] and [7].

An example in [12] shows that the above definition of entropy is indeed anatural generalization of the classical one in Ergodic Theory. Later we willconsider again this example when analyzing Lyapunov exponents (see section8).

4 Process Xn, n ∈ N, taking values on P (Ck)

Consider a fixed measure µ on Mk and a fixed L : Mk → Mk, such that,∫Mk‖L(v)‖2 dµ(v) <∞, and, also assume that φL is irreducible and stochas-

tic.Note that if, for example, µ is a probability and the the function v →

‖L(v)‖ is bounded we get that∫Mk‖L(v)‖2 dµ(v) <∞.

We follow the notation of [10] (and, also [12])Denote by P (Ck) the projective space on Ck with the metric d(x, y) =

(1 − |〈x , y 〉|2)1/2, where x, y are representatives with norm 1 and 〈 · , · 〉 isthe canonical inner product.

Take x ∈ P (Ck) and S ⊂ P (Ck). For a stochastic φL we consider thekernel

ΠL(x, S) =

∫Mk

1S(L(v) · x) ‖L(v)x‖2 dµ(v), (1)

where the norm above is the Hilbert-Schmidt one.This discrete time process (described by the kernel) taking values on

P (Ck) is determined by such µ and L. If ν is a probability on the Borelσ-algebra B of P (Ck) define

νΠL(S) =

∫P (Ck)

ΠL(x, S) dν(x)

=

∫P (Ck)×Mk

1S(L(v) · x) ‖L(v)x‖2 dν(x)dµ(v).

7

νΠL is a new probability on P (Ck) and ΠL is a Markov operator. Theabove definition of ν → νΠL is a simple generalization of the one in [10],where the authors take the L consider here as the identity transformation.

The map ν → ν ΠL (acting on probabilities ν) is called the Markov op-erator obtained from φL in the paper [21] . There the a priori measure µ isa sum of Dirac probabilities. Here we consider a more general setting.

Definition 4.1 We say that the probability ν over P (Ck) is invariant forΠL, if νΠL = ν.

See [12] (and also future Proposition 6.7) for the existence of invariantprobabilities for ΠL

5 The process ρn, n ∈ N, taking values on Dk

For a fixed µ over Mk and L such φL is irreducible and stochastic, one cannaturally define a process (ρn) on Dk = ρ ∈ Mk : tr ρ = 1 and ρ ≥ 0,which is called quantum trajectory.

We follow the notation of [10] (and, also [12])Given a ρ0 initial state, we get

ρn =L(v)ρn−1L(v)∗

tr (L(v)ρn−1L(v)∗)

with probabilitytr (L(v)ρn−1L(v)∗)dµ(v), n ∈ N.

We want to relate the invariant probabilities of last section with the fixedpoint ρinv = ρLinv of φL.

First, denote Ω := MNk , and for ω = (ωi)i∈N, take φn(ω) = (ω1, ..., ωn).

We denote πn the projection of ω in its first n coordinates.We also denote by M the Borel sigma algebra Mk. For all, n ∈ N,

consider On the sigma algebra on Ω generated by the cylinder sets of sizen, that is, On := π−1

n (Mn). We equip Ω with the smaller sigma algebra Owhich contains all On, n ∈ N.

Denote Jn := B ⊗ On and J := B ⊗ O. In this way, (P (Ck) × Ω,J )is an integrable space. By abuse of language we consider Vi : Ω → Mk as arandom variable Vi(ω) = ωi. We also introduce another random variable

Wn := L(Vn)...L(V1), where Wn(ω) = L(ωn)...L(ω1).

8

We point out that the symbol ⊗ does not represents tensor product.For a given a probability ν on P (Ck), we define for S ∈ B and On ∈ On,

another probability

Pν(S ×On) :=

∫S×On

‖Wn(ω)x‖2 dν(x)dµ⊗n(ω).

Denote Eν the expected value with respect to Pν . Now observe that fora ν probability on P (Ck), if πX0 : Ck − 0 → P (Ck) is the orthogonalprojection on subspace generated by X0 on Ck, we have

ρν := Eν(πX0) =

∫P (Ck)

πx0 dν(x0).

We call ρν barycenter of ν, and it’s easy to see that ρν ∈ Dk.

Proposition 5.1 If ν is invariant for ΠL, then

ρν = Eν(πX0) = Eν(πX1

) = φL(ρν).

Therefore, for an irreducible L, every invariant measure ν for ΠL has thesame barycenter.

We point out that in this way we can recover ρinv, the fixed point of φL,by taking the barycenter of any invariant probability (the quantum channelφL admits only one fixed point). That is, for any invariant probability ν forΠL, we get that ρν = ρinv.

Note that the previous process can be seen as ρn : Ω → Dk, such that,ρ0(x, ω) = ρν and

ρn(ω) =Wn(ω)ρ0Wn(ω)∗

tr (Wn(ω)ρ0Wn(ω)∗).

Using an invariant ρ we can define an Stationary Stochastic Process takingvalues on Mk. That is, we will define a probability P over Ω = (Mk)

N.Take On ∈ On and define

Pρ(On) =

∫On

tr (Wn(ω)ρWn(ω)∗) dµ⊗n(ω).

The probability P on Ω defines a Stationary Stochastic Process.

9

6 Irreducibility, the Φ-Erg property and the

purification condition

We will use in this section the notation of [10].

Definition 6.1 Given L : Mk → Mk, µ on Mk and E subspace of Ck, wesay that E is (L, µ)-invariant, if L(v)E ⊂ E, for all v ∈ suppµ.

Definition 6.2 Given L : Mk → Mk, µ on Mk, we say that L is Φ-Ergfor µ, if there exists an unique minimal non-trivial space E, such that, E is(L, µ)-invariant.

In [25] it is shown that if the above space E is equal to Ck, then L isirreducible for µ (or, µ-irreducible) in the sense of Definition 2.11.

The relation of Pρ and Pν (described on last sections) is described in thenext result.

Proposition 6.3 The marginal of Pν on O is Pρν . In the case the Φ-Erg istrue, then for any two Π-invariant probabilities νa and νb, we get Pρνa = Pρνb .

The proof of the above result when L is the identity was done in Propo-sition 2.1 in [10]. The proof for the case of a general L is analogous.

Given two operators A and B we say that A ∝ B, if there exists β ∈ C,such that, A = β B.

Definition 6.4 Given L : Mk → Mk, µ on Mk, we say that the pair (L, µ)satisfies the purification condition, if an orthogonal projector π, such that,for any n ∈ N

πL(V1)∗...L(Vn)∗ L(Vn)...L(V1)π ∝ π,

for µ⊗n-almost all (v1, v2, .., vn), it is necessarily of rank one.

Following [10] we denote Pch = P 1kId.

We denote by Yn, n ∈ N, the matrix-valued random variable

Yn =W ∗nWn

tr (W ∗nWn)

, if tr (W ∗nWn) 6= 0,

where we extend the definition in arbitrary way when tr (W ∗nWn)=0 .

The next two propositions are of fundamental importance in the theoryand they were proved in Proposition 2.2 in [10] (the same proofs works inour setting).

10

Proposition 6.5 For any probability ν over P (Ck) the stochastic processYn, n ∈ N, is a martingale with respect to the sequence of sigma-algebras On,n ∈ N. Therefore, there exists a random variable Y∞ which is the almostsure limit of Yn for the probability Pν and also in the L1 norm.

Proposition 6.6 For any probability ν over P (Ck) and ρ ∈ Dk

dPρ

dPch= k tr (ρ Y∞).

Moreover, µ and L satisfy the purification condition, if and only if, Y∞is Pν-a.s a rank one projection for any probability ν over P (Ck).

Proposition 6.7 If the pair (L, µ) satisfies the φ-Erg and the purificationcondition, then, the Markov kernel Π admits a unique invariant probability.

x1 ∧ x2 ∧ ... ∧ xn, with xj ∈ Ck, denotes the classical wedge product (analternate form on Ck).

One can consider an inner product

〈 r1 ∧ r2 ∧ ... ∧ rn , s1 ∧ s2 ∧ ... ∧ sn 〉 = det (risj))i,j=1,2,...,n,

and, the associated norm |x1 ∧ x2 ∧ ... ∧ xn |.Given an operator X : Ck → Ck we define

∧nX : ∧nCk → ∧nCk by∧nX (x1 ∧ x2 ∧ ... ∧ xn) = X(x1) ∧X(x2) ∧ ... ∧X(xn).

Proposition 6.8 Assume the pair (L, µ) satisfies the purification condition,then, there are two constants C > 0 and β < 1, such that, for each n∫Mnk

| ∧2 (L(vn)...L(v1) ) | d µ⊗n(v1, v2, ..., vn) = Ech (k|∧2 Wn |

tr (W ∗nWn )

) ≤ C βn.

The proofs of the two propositions above are similar to the correspondingones in [10].

Consider B(Mk) = L : Mk → Mk |L is continuous and bounded where‖L‖ = supv∈Mk

‖L(v)‖.

Definition 6.9 For a fixed a measure µ over Mk, define

Bµ(Mk) = L ∈ B |L is µ-irreducible,

and

BΦµ (Mk) = L ∈ B |L is Φ-Erg for µ.

11

Proposition 6.10 Given µ over Mk with #suppµ > 1, BΦµ (Mk) is open and

dense on B(Mk).

Proof: See [12].

In section 9 we will prove:

Proposition 6.11 Given µ over Mk with #suppµ > 1, the set of L satisfy-ing the purification condition is generic in B(Mk).

7 Lyapunov exponents for quantum channels

In this section we will consider a discrete time process taking values on Mk.Take µ over Mk and L : Mk → Mk in such way that the associated

channel Φ defines a Φ-Erg stochastic map. We assume in this section thatρ ∈ Dk is such that Φ(ρ) = ρ. Such ρ plays the role of the initial vector ofprobability (in the analogy with the theory of Markov Chains).

We follow the notation of [10].Take Ω = MN

k , and for n ∈ N let On be the σ-algebra on Ω generated bythe n-cylinder sets (as in Section 5).

An element on Ω is denoted by (ω1, ω2, ..., ωn, ...). Following section 5 wedenote Wn(ω) = L(ωn)...L(ω1).

Taking On ∈ On we define

P(On) =

∫On

tr (Wn(ω)ρWn(ω)∗) dµ⊗n(ω).

If O is the smallest σ-algebra of Ω that contains all On we can extendthe action of P to this σ-algebra.

The probability P on Ω defines a Stationary Stochastic Process.

Theorem 7.1 (Ω,P, θ) is ergodic where θ is the shift map.

The above theorem has been proved in Lemma 4.2 in [10].

Theorem 7.2 Suppose the pair (L, µ) satisfies irreducibility, the φ-Erg andthe purification condition. Assume also that

∫|L(v)|2 log |L(v)| dµ(v) <∞,

then, there exists numbers

∞ > γ1 ≥ γ2 ≥ ... ≥ γk ≥ −∞,

12

such that, for any probability ν over P (Ck) and any p ∈ 1, 2, .., k

limn→∞

1

nlog |

p∧Wn | =

p∑j=1

γj,

Pν-a.s.

The above theorem was proved in [10] and the same proof works here inour setting. We point out that a key ingredient in this proof (see (35) in [10])is the fact that if (L, µ) is φ-Erg and irreducible, then, ρinv > 0, and for anyρ ∈ Dk

Pρ << Pρinv .

Proposition 6.3 is also used in the proof (corresponds to proposition 2.1in [10]).

The numbersγ1 ≥ γ2 ≥ ... ≥ γk,

are called the Lyapunov exponents.

Theorem 7.3 Suppose that L is generic for µ and∫|L(v)|2 log |L(v)| dµ(v) <

∞, then, the Lyapunov exponents

∞ > γ1 ≥ γ2 ≥ ... ≥ γk ≥ −∞

are well defined.(a) γ2 − γ1 < 0, where γ2 − γ1 is the limit

limn→∞

1

nlog|∧2Wn ||Wn |2

,

whenever γ1 = −∞

(b) For Pν-almost sure x we have that

limn→∞

1

n(log |Wn (x) | − log |Wn |) = 0.

proof:The proof of (a) is similar to the one in [10].

Proof of (b): We start with

‖Wnx‖‖Wn‖

=[tr (WnπxW

†n) ]1/2

[tr (WnW†n) ]1/2

=

[tr (W †

nWnπx)

tr (WnW†n)

]1/2

13

=

[tr

(W †nWn

tr (WnW†n)

πx

) ]1/2

= [tr (Mnπx)]1/2 .

These calculations are valid for all ω such that WnW†n(ω) 6= 0. Since

Pν(Wn = 0) = 0, no extra work is required. Now we use proposition 2.2 in[10] which says that Mn converges Pν−a.s. and in L1 norm to aO-measurablerandom variable M∞. By continuity of the trace and square root, we have

limn→∞

[tr (Mnπx)]1/2 = [tr (M∞πx)]

1/2 , for Pν − a.s. x ∈ P (Ck).

The proof is similar to the one in [10].

8 The main example - Lyapunov exponents

and entropy

Now we will present an example where we can estimate the Lyapunov expo-nents and show a relation with entropy.

Let Vij =√pij| i 〉〈 j | where P = (pij) is a irreducible (in the classical

sense for a Markov chain) k by k column stochastic matrix, µ =∑

ij δVij andL = I. In this case, we get that

V ∗ijVij = pij| j 〉〈 j |,

is a diagonal matrix and, if A = (aij),

V ∗ijAVij = pijaii| j 〉〈 j |.Therefore, when ω = (Vinjn), we have

W2(ω)∗W2(ω) = pi1j1pi2j2| j2 〉〈 j2 |δi2j1 ,where δij = 0 if i 6= j and 1 if i = j. By induction,

Wn(ω)∗Wn(ω) =

(n∏k=1

pikjk

)(n−1∏k=1

δik+1jk

)| jn 〉〈 jn |.

Thus, Wn(ω)∗Wn(ω) is 0 or a diagonal matrix with a unique entry differentfrom 0. This entry is exactly (

∏nk=1 pikjk) which implies that

‖Wn(ω)∗Wn(ω)‖ =

(n∏k=1

pikjk

)= p

X11,n(ω)11 p

X12,n(ω)12 p

X21,n(ω)21 p

X22,n(ω)22 ,

14

where

Xij,n(ω) =n−1∑k=0

1[Vij ] θk(ω),

with 1[Vij ] being the characteristic function of cylinder [Vij].Note that under the ergodicity hypothesis we would have the property:

for any i, j and P-almost sure ω, we have that

limn→∞

1

n

n−1∑k=0

1[Vij ] θk(ω) = P([Vij]).

It follows from the arguments in example 8.5 in [12] that the pair (L, µ)satisfies the purification condition, the φ-Erg conditions and also irreduciblil-ity.

Remember that for given matrix A ∈ Mk(C), a1(A) ≥ a2(A) ≥ ... ≥ak(A) are the singular values of A, i.e., the square roots of eigenvalues ofA∗A, labeled in decreasing order. From Lemma III.5.3 in [11] we havewwwww

p∧Wn(ω)

wwwww = a1(Wn)...ap(Wn).

Therefore,ww∧1Wn(ω)

ww = a1(Wn) = ‖W ∗nWn‖

12 .

Following Proposition 7.2 (which corresponds to Proposition 4.3 in [10])we can obtain the greater Lyapunov exponent γ1 taking the limit

γ1 : = limn→∞

1

n

wwwww1∧Wn(ω)

wwwww= lim

n→∞

1

nlog(‖Wn(ω)∗Wn(ω)‖

12 )

=1

2

∑ij

limn→∞

Xij,n(ω)

nlog(pij)

=∑ij

P([Vij])

2log(pij).

We know from [12] that if Φ is the channel defined for such µ and L,then, Φ is Φ-Erg. Moreover, the unique ρ, such that, Φ(ρ) = ρ, is exactlythe diagonal matrix ρ with entries π1, π2, ..., πk, where π = (π1, π2, ..., πk) isthe invariant probability vector for the stochastic matrix P . We also know

15

in this case that the entropy (see example 8.5 in [12]) for a channel defined in[12] is equal to the classical Shannon-Kolmogorov entropy of the stationaryMarkov Process associated to the column stochastic matrix P = (pij) (seeformula in [24]).

Now, we can estimate

P([Vij]) =

∫[Vij ]

tr (vρv∗) dµ(v) = tr (V ∗ijVijρ) = tr (pij| j 〉〈 j |ρ) = pij〈 j |ρ| j 〉

= pijπj.

Therefore,

γ1 =1

2

∑i,j∈0,1

πjpij log(pij) = −1

2h,

where h is the entropy of the Markov invariant measure associated to thematrix P .

The value 12

which multiplies the entropy on the above expression is dueto the fact that we considered the norm ‖A‖ = 〈A , A 〉1/2.

Now we estimate the second Lyapunov exponent γ2.We showed that Wn(ω)∗Wn(ω) = (

∏nk=1 pikjk)

(∏n−1k=1 δik+1jk

)| jn 〉〈 jn |,

which implies that the second eigenvalue is 0 and therefore a2(Wn(ω)) = 0.Now, we can get γ2, indeed,

γ1 + γ2 = limn

1

nlog(a1(Wn(ω))a2(Wn(ω))) = lim

n

1

nlog(0),

which implies that γ2 = −∞.

9 The purification condition is generic

The measure µ is fixed from now on.Our main goal in this section is to show:

Proposition 9.1 Given µ over Mk with #suppµ > 1, the set of L satisfyingthe purification condition is generic in B(Mk).

This will follow from Lemma 9.14.

Definition 9.2 We say that the projection π n-purifies L : Mk →Mk, whererank π ≥ 2, if there exists E ∈ On, with µ⊗n(E) > 0, such that,

πWn(ω)Wn(ω)π 6∝ π,

for all ω ∈ E.

16

In order show that a certain L satisfies the purification condition we haveto consider all possible projections π (see definition 6.4).

Observe that if Q is a unitary matrix, π has rank great or equal to 2 andn-purifies L for E ∈ On, then

QπQ∗QWn(ω)∗Wn(ω)Q∗QπQ∗ 6∝ QπQ∗.

Besides that, Wn(ω) = L(ωn)...L(ω1), so, as Q∗Q = Idk, we have

QWn(ω)∗Wn(ω)Q∗ =

QL(ω1)∗Q∗Q...Q∗QL(ωn)∗Q∗QL(ωn)Q∗Q...Q∗QL(ω1)Q∗Q.

From this follows:

Proposition 9.3 If LQ(v) := QL(v)Q∗, then for a projection π, such that,rank π ≥ 2 and an unitary matrix Q, it’s true that

π n-purifies L ⇐⇒ QπQ∗ n-purifies LQ.

Definition 9.4 For an orthogonal projection π and n ∈ N we define

Purnπ = L ∈ B(Mk) |π n-purifies L.

Note that if

Pur = L ∈ B(Mk) |ΦL satisfies (Pur) condition ,

and we denote

P2 = π orthogonal projection | rankπ ≥ 2,

it follows thatPur =

⋂π∈P2

⋃n∈N

Purnπ .

Proposition 9.5 For any π ∈ P2 and n ∈ N, Purnπ is open.

proof: Take π ∈ P2 with rank π = l, Q an unitary matrix that diagonalizesπ. Suppose that

17

π := QπQ∗ =

1 0 . . . 0 0 . . . 00 1 . . . 0 0 . . . 0...

.... . .

......

. . ....

0 0 . . . 1 0 . . . 00 0 . . . 0 0 . . . 0...

.... . .

......

. . ....

0 0 . . . . . . 0 . . . 0

.

So, if L ∈ Purnπ , we know that π n-purifies LQ.If

(sLij(ω)) := QWn(ω)∗Wn(ω)Q∗

we know that sLij are continuous functions. Moreover, there exists ω0 ∈suppµ⊗n such that at least one of following conditions occurs:

1. There exists i 6= j, such that, sLij(ω0) 6= 0, or there exists j = i > rankπ, such that, sLii(ω0) 6= 0.

In this case, we define the matrix sL(ω) := sLij(ω);

2. There exists i 6= j, such that, sLii(ω0)− sLjj(ω0) 6= 0.In this case, we define sL(ω) := sLii(ω)− sLjj(ω).

Of course that for F ∈ B(Mk) with ‖L− F‖ ≤ ε, we have, for ε smallenough, that sF (ω0) 6= 0, because sF (ω0) has a continuous dependence on F.Furthermore, sF is continuous, then exists a open set B with ω0 ∈ B, suchthat, sF (ω) 6= 0, for ω ∈ B. Moreover, ω0 ∈ suppµ⊗n which implies thatµ⊗n(B) > 0, and therefore F ∈ Purnπ .

We point out that sL(w0) 6= 0 is the good condition for purification.

Proposition 9.6 For any π ∈ P2, Pur1π is dense.

proof: Take L /∈ Pur1π and Q unitary matrix that diagonalizes π as above.

If L /∈ Pur1π, defining (sLij) as in the previous proposition, we know that

s11(v) = s22(v), for almost every v. If D = | 1 〉〈 1 |, and ε > 0, we considerLεQ(v) = LQ(v) + εD. So, we have

LεQ(v)∗LεQ(v) = (LQ(v) + εD)∗(LQ(v) + εD)

= LQ(v)∗LQ(v) + εL∗Q(v)D + εDLQ(v) + ε2D.

18

This perturbation change sL11(v) but not sL22(v), which implies that QπQ∗

n-purifies LεQ. Now, we need just to take Lε(v) = L(v) + εQ∗DQ and, as‖L− Lε‖ is small, it follows the density property at once.

Definition 9.7 For any π ∈ P2, we define Purπ =⋃n∈N Pur

nπ .

Note that by the two propositions above given a fixed π the set Purπ isan open and dense on B(Mk). The purification condition requires to considerall possible projections π (see definition 6.4).

Lemma 9.8 If π1, π2 ∈ P2 has the same rank, Ei := Im πi, and xi is anorthonormal basis for Ei, then, π2xi is a basis for E2 if π1 and π2 are closeenough.

proof: The proof will be by contradiction. Suppose rankπ1 = l and yi :=π2xi are linearly dependent. So, the dimension generated by yi is at mostl−1. Then, there exists a vector y ∈ E2 which has norm 1 and it is orthogonalto the subspace generated by yi. Therefore, 〈 y , yi 〉 = 0, for all i. Thisimplies that 〈 y , π2xi 〉 = 〈π2y , xi 〉 = 〈 y , xi 〉 = 0 and moreover π1y = 0.Finally, we get ‖π1 − π2‖ ≥ ‖π1y − π2y‖ = ‖y‖ = 1.

If we assume that ‖π1 − π2‖ < 1 we are done.

Observe that, for i 6= j and ‖π2 − π1‖ < ε,

|〈 yi , yj 〉| = |〈 yi , xj 〉| = |〈 yi − xi , xj 〉| ≤ ‖yi − xi‖ ‖xj‖ = ‖yi − xi‖

≤ ‖π2 − π1‖ < ε (2)

The set of yi is not an orthonormal basis.We would like to get an orthonormal basis close to the orthogonal basis

x1, .., xn. Our aim is to prove corollary 9.12 which claims that, given ε thereexists an orthonormal basis (ui) for E2 with ‖ui − xi‖ < Cε, for some con-stant C > 0. In this direction we will perform a Gram-Schmidt normalizationprocedure.

Denote u1 :=y1

‖y1‖, Ni := yi−

∑i−1j=1〈 yi , uj 〉uj and ui :=

Ni

‖Ni‖, for i > 1.

Then, we have

19

‖u1 − x1‖ =

wwww ‖π2x1‖−1 π2x1 − ‖π2x1‖−1 x1 + ‖π2x1‖−1 x1 − x1

wwww≤ ‖π2x1‖−1 ‖π2x1 − π1x1‖+ ‖x1‖

‖π2x1‖−1 − 1

≤ ‖π2x1‖−1 ε+‖π2x1‖−1 − 1

< ε(1− ε)−1 + (1− ε)−1ε

< 4ε,

and

|‖π2xi‖ − ‖xi‖| ≤ ‖π2xi − π1xi‖ < ε =⇒ 1− ε < ‖π2xi‖ < 1 + ε.

Furthermore,

‖y2 − 〈 y2 , u1 〉u1‖ =wwy2 − ‖π2xi‖−2 〈 y2 , y1 〉y1

ww≤ ‖x2‖+ (1− ε)−2ε ‖x1‖< 1 + 4ε.

Proposition 9.9 For any i ∈ 1, ..., n, j < i, there is Cij > 0, such that,|〈 yi , uj 〉| < Cijε.

proof:Take N := min ‖Ni‖ > 0.Observe that |〈 y2 , u1 〉| ≤ N−1ε (this follows from a similar procedure as

in (2) and the Cauchy-Schwartz inequality).Then suppose, for all l < i, |〈 yl , uj 〉| ≤ Cljε, for all j < l, with Clj > 0.

If j < i, we have

|〈 yi , uj 〉| ≤ N−1 |〈 yi , yj 〉|+N−1

j−1∑k=1

|〈 yj , uk 〉|

≤ N−1ε+N−1

j−1∑k=1

Cjkε

=

(1 +

j−1∑k=1

Cjk

)N−1ε.

Taking Cij =

(1 +

∑j−1k=1Cjk

)N−1 the claim follows by induction.

20

Proposition 9.10 For all i, |‖Ni‖ − 1| < Kε, for some K > 0.

proof:

‖Ni‖ ≤ 1 + ‖yi − xi‖+i−1∑j=1

|〈 yi , uj 〉|

≤ 1 + ε+

( i−1∑j=1

Cij∑)

ε

= 1 +

(1 +

( i−1∑j=1

Cij

))ε.

Taking K =1 +

∑i−1j=1 Cij

the proof is done.

Proposition 9.11 For all i, we have ‖ui − xi‖ < Ciε.

proof:

‖ui − xi‖ ≤ N−1 ‖Ni − xi‖+ ‖‖Ni‖xi − xi‖

≤ N−1 ‖yi − xi‖+N−1

i−1∑j=0

|〈 yi , uj 〉|+ |‖Ni‖ − 1|

≤ N−1ε+N−1

i−1∑j=1

Cij +Kε

=

(N−1 +N−1

i−1∑j=1

Cij +K

)ε.

Define Ci := N−1+N−1∑i−1

j=1Cij+K and the statement has been proved.

Corollary 9.12 There exists C > 0, such that, for all ε > 0, there exists anorthonormal basis (ui) for E2 with ‖ui − xi‖ < Cε.

If we repeat the process, but now for E⊥1 and E⊥2 , we get another constantC2, and in a similar way we obtain new vectors (ui) from the (xi). These uidefine an orthonormal basis for Ck with ‖ui − xi‖ < C2ε.

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Now we define Q1, Q2 , such that Q1xi = ei and Q2ui = ei, where (ei)is canonical basis for Ck. Then, QiπiQ

∗i is a diagonalization for πi. Observe

that‖Q2Q

∗1 − I‖ = ‖Q2Q

∗1 −Q2Q

∗2‖ < C3ε,

the map A → Q2Q∗1A(Q2Q

∗1)∗ is continuous and moreover this map is close

to the identity map.

We know if L ∈ Purnπ1 , then, we take sL1 from the π1, Q1 and L, as inproposition 9.5. In the same way there exists ω0 ∈ suppµ with sL1 (ω0) 6= 0.Observe that sL1 (ω0) depends only on the cordinates ofQ1W

Ln (ω0)∗WL

n (ω0)Q∗1.Now, applying Q2Q

∗1 (·) (Q2Q

∗1)∗ we get Q2W

Ln (ω0)∗WL

n (ω0)Q∗2. Note thatthis is the same as to consider π2, Q2, L and the associated sL2 (ω0). If ε issmall enough, sL2 (ω0) 6= 0 and we can repeat the argument used in proposi-tion 9.5 in order to obtain an open set B, such that, ω0 ∈ B and, moreover,if ω ∈ B then sL2 (ω) 6= 0. Therefore, L ∈ Purnπ2 .

The previous arguments prove the following lemma.

Lemma 9.13 If L ∈ Purπ1 and π1, π2 are close enough, then L ∈ Purπ2.

Lemma 9.14 Take K2 a countable dense subset of P2. Then,

Pur =⋂π∈K2

Purπ.

proof: We will use the classical Baire Theorem. Suppose that L ∈⋂π∈K2

Purπ,then for lemma 9.13, for each π ∈ K2, as L ∈ Purπ, there exists ε(π), suchthat, if π ∈ P2 and ‖π − π‖ < ε(π), then L ∈ Purπ. Furthermore, if we defineB(π) = π ∈ P2| ‖π − π‖ < ε(π), then

⋃π∈K2

B(π) covers P2. Therefore,for any π ∈ P2, there is π ∈ K2, such that, π ∈ B(π) and thus L ∈ Purπ.This implies that L ∈

⋂π∈P2

Purπ = Pur.

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